> with(DEtools):
phug:=[ D(theta)(t) = v(t) - cos(theta(t))/v(t),
D(v)(t)
= -sin(theta(t)) -R*v(t)^2];
(1)
> R:=0.3;
s t u f f : = [ t h e t a ( t ) , v ( t ) ] , t = - 1 . . 2 0,
theta=-Pi..2*Pi, v=0..3,
[ s e q ( [ t h e t a ( 0 ) = 0 , v ( 0 ) = i ] , i =1 . . 3, 0 . 2 ) ] ,
linecolor=[seq(COLOR(HUE,i),i=0..1,.1)], stepsize=0.05:
DEplot(phug, stuff, scene=[theta,v]);
3
2
v
1
0
1
2
3
4
theta
5
6
> R:=0.3;
s t u f f : = [ t h e t a ( t ) , v ( t ) ] , t = - 1 . . 2 0,
theta=1..2, v=0..0.1,
[seq([theta(0)=Pi/2+i, v(0)=0.1],i=-0.1..0.1,0.01)],
linecolor=[seq(COLOR(HUE,i),i=-0.1..0.1,0.01)], stepsize=0.05:
DEplot(phug, stuff, scene=[theta,v]);
v
0
1
2
theta
> s t u f f : = [ t h e t a ( t ) , v ( t ) ] , t = - 1 . . 2 0,
theta=1..2, v=0..0.1,
[seq([theta(0)=Pi/2+i, v(0)=0.1],i=-0.1..0.1,0.01)],
l i n e c o l o r = [ s e q ( C O L O R ( H U E , i ) , i = - 0 . 1 . . 0 . 1 , 0 . 0 1 ) ] , s t e p s i z e = 0 . 0 0 5:
DEplot(phug, stuff, scene=[theta,v]);
v
0
1
2
theta
> s t u f f : = [ t h e t a ( t ) , v ( t ) ] , t = - 1 . . 2 0,
theta=-Pi..2*Pi, v=0..3,
[ s e q ( [ t h e t a ( 0 ) = 0 , v ( 0 ) = i ] , i =1 . 8 . . 2, 0 . 0 5 ) ] ,
l i n e c o l o r = [ s e q ( C O L O R ( H U E , i ) , i =1 . 8 . . 2, 0 . 0 5 ) ] , s t e p s i z e = 0 . 0 5 :
DEplot(phug, stuff, scene=[theta,v]);
3
2
v
1
0
1
2
3
4
theta
5
6
> s t u f f : = [ t h e t a ( t ) , v ( t ) ] , t = - 5 . . 1 0,
theta=-0..Pi, v=0..2,
[ [ t h e t a ( 0 ) = P i / 2 , v ( 0 ) = 0 . 0 1 ] , s e q ( [ t h e t a ( 0 ) = 0 , v ( 0 ) = i ] , i =1 . 8 . . 2,
0.05)],
l i n e c o l o r = [ b l a c k , s e q ( C O L O R ( H U E , i ) , i =1 . 8 . . 2, 0 . 0 5 ) ] , o b s r a n g e =
false,stepsize=0.05:
DEplot(phug, stuff, scene=[theta,v]);
2
v
1
0
1
2
3
theta
> s t u f f : = [ t h e t a ( t ) , v ( t ) ] , t = - 5 . . 1 0,
theta=-0..Pi, v=0..2,
[ [ t h e t a ( 0 ) = P i / 2 , v ( 0 ) = 0 . 0 1 ] , s e q ( [ t h e t a ( 0 ) = 0 , v ( 0 ) = i ] , i =1 . 8 . . 2,
0.05)],
l i n e c o l o r = [ b l a c k , s e q ( C O L O R ( H U E , i ) , i =1 . 8 . . 2, 0 . 0 5 ) ] , o b s r a n g e =
false,stepsize=0.005:
DEplot(phug, stuff, scene=[theta,v]);
2
v
1
0
1
2
3
theta
> s t u f f : = [ t h e t a ( t ) , v ( t ) ] , t = - 5 . . 1 0,
theta=-0..Pi, v=0..2,
[ [ t h e t a ( 0 ) = P i / 2 , v ( 0 ) = 0 ] , s e q ( [ t h e t a ( 0 ) = 0 , v ( 0 ) = i ] , i =1 . 8 . . 2,
0.05)],
l i n e c o l o r = [ b l a c k , s e q ( C O L O R ( H U E , i ) , i =1 . 8 . . 2, 0 . 0 5 ) ] , o b s r a n g e =
false,stepsize=0.005:
DEplot(phug, stuff, scene=[theta,v]);
Warning, plot may be incomplete, the following errors(s) were
issued:
cannot evaluate the solution past the initial point, problem
may be complex, initially singular or improperly set up
2
v
1
0
1
2
3
theta
> s t u f f : = [ t h e t a ( t ) , v ( t ) ] , t = - 5 . . 1 0,
theta=1.5..1.8, v=0..0.2,
[[theta(0)=Pi/2, v(0)=0.1],[theta(0)=Pi/2,v(0)=0.01]],
l i n e c o l o r = [ b l a c k , s e q ( C O L O R ( H U E , i ) , i =1 . 8 . . 2, 0 . 0 5 ) ] , o b s r a n g e =
false,stepsize=0.005:
DEplot(phug, stuff, scene=[theta,v]);
v
0
theta
> vphug:=[ D(theta)(t) = v(t)^2 - cos(theta(t)),
D(v)(t)
= (-sin(theta(t)) -R*v(t)^2)*v(t)];
(2)
> s t u f f : = [ t h e t a ( t ) , v ( t ) ] , t = - 5 . . 1 0,
theta=-Pi/2..Pi, v=0..2,
[ [ t h e t a ( 0 ) = P i / 2 , v ( 0 ) = 0 . 0 1 ] , s e q ( [ t h e t a ( 0 ) = 0 , v ( 0 ) = i ] , i =1 . 8 . . 2,
0.05)],
l i n e c o l o r = [ b l a c k , s e q ( C O L O R ( H U E , i ) , i =1 . 8 . . 2, 0 . 0 5 ) ] , o b s r a n g e =
false,stepsize=0.05:
DEplot(vphug, stuff, scene=[theta,v]);
Warning, plot may be incomplete, the following errors(s) were
issued:
cannot evaluate the solution further left of -.69973011,
probably a singularity
Warning, plot may be incomplete, the following errors(s) were
issued:
cannot evaluate the solution further left of -.64930540,
probably a singularity
Warning, plot may be incomplete, the following errors(s) were
issued:
cannot evaluate the solution further left of -.60461454,
probably a singularity
Warning,
issued:
cannot
probably
Warning,
issued:
cannot
probably
plot may be incomplete, the following errors(s) were
evaluate the solution further left of -.56476163,
a singularity
plot may be incomplete, the following errors(s) were
evaluate the solution further left of -.52902775,
a singularity
2
v
1
0
1
2
3
theta
> F:=(theta,v)-> [ v^2-cos(theta), v*(-sin(theta)-S*v^2)];
(3)
> with(VectorCalculus):
> Jacobian(F(theta,v), [theta,v]);
(4)
> eval(%, {theta=Pi/2, v=0});
(5)
> Jacobian(F(theta,v), [theta,v]):
(6)
>
eval(%, {theta=-Pi/2, v=0});
(6)
Want plane to stop when y(t)=0.
> xphug:=[ D(theta)(t) = v(t) - cos(theta(t))/v(t),
D(v)(t)
= -sin(theta(t)) -R*v(t)^2,
D(x)(t)
= v(t)*cos(theta),
D(y)(t)
= v(t)*sin(theta)];
(7)
> crashy:=[ D(theta)(t) = v(t) - cos(theta(t))/v(t),
D(v)(t)
= -sin(theta(t)) -R*v(t)^2,
D(x)(t)
= v(t)*cos(theta(t)),
D(y)(t)
= piecewise( y(t)>0, v(t)*sin(theta(t)), 0)];
(8)
> with(plots):
> R:=1;
s t u f f : = [ t h e t a ( t ) , v ( t ) , x ( t ) , y ( t ) ] , t = - 1 . . 2 0,
theta=-Pi..4*Pi, v=0..3, x=-3..10, y=-2..3,
[ s e q ( [ t h e t a ( 0 ) = 0 , v ( 0 ) = i , x ( 0 ) = 0 , y ( 0 ) = 0 ] , i =1 . . 3, 0 . 2 ) ] ,
linecolor=[seq(COLOR(HUE,i),i=0..1,.1)], stepsize=0.05:
display( array( [ DEplot(crashy, stuff, scene=[theta,v]),
DEplot(crashy, stuff, scene=[x,y]) ]));
3
3
2
y
2
1
v
1
0
2
4
6
8
10
x
0
2
4
6
8
theta
10
12
> crashy:=[ D(theta)(t) = piecewise( y(t)>0,v(t) - cos(theta(t))/v
(t),0),
D(v)(t)
= piecewise( y(t)>0,-sin(theta(t)) -R*v(t)^2,
0),
D(x)(t)
= piecewise( y(t)>0,v(t)*cos(theta(t)),0),
D(y)(t)
= piecewise( y(t)>0, v(t)*sin(theta(t)),0)];
(9)
> R:=1;
s t u f f : = [ t h e t a ( t ) , v ( t ) , x ( t ) , y ( t ) ] , t = - 1 . . 2 0,
theta=-Pi..4*Pi, v=0..3, x=-3..10, y=-2..3,
[ s e q ( [ t h e t a ( 0 ) = 0 , v ( 0 ) = i , x ( 0 ) = 0 , y ( 0 ) = 0 ] , i =1 . . 3, 0 . 2 ) ] ,
linecolor=[seq(COLOR(HUE,i),i=0..1,.1)], stepsize=0.05:
display( array( [ DEplot(crashy, stuff, scene=[theta,v]),
DEplot(crashy, stuff, scene=[x,y]) ]));
3
3
2
y
2
1
v
1
0
2
4
6
8
x
0
2
4
6
8
theta
10
12
> R:=0.5;
s t u f f : = [ t h e t a ( t ) , v ( t ) , x ( t ) , y ( t ) ] , t = - 1 . . 2 0,
theta=-Pi..4*Pi, v=0..3, x=-3..10, y=-2..3,
[ s e q ( [ t h e t a ( 0 ) = 0 , v ( 0 ) = i , x ( 0 ) = 0 , y ( 0 ) = 1 ] , i =1 . . 3, 0 . 2 ) ] ,
linecolor=[seq(COLOR(HUE,i),i=0..1,.1)], stepsize=0.05:
display( array( [ DEplot(crashy, stuff, scene=[theta,v]),
DEplot(crashy, stuff, scene=[x,y]) ]));
10
3
3
2
y
2
1
v
1
0
2
4
6
x
0
2
4
6
8
theta
10
12
8
10